mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, an orthodox semigroup is a

regular semigroup In mathematics, a regular semigroup is a semigroup ''S'' in which every element is regular, i.e., for each element ''a'' in ''S'' there exists an element ''x'' in ''S'' such that . Regular semigroups are one of the most-studied classes of semigroup ...

whose set of

idempotent Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...

s forms a

subsemigroup In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively (just notation, not necessarily the ...

. In more recent terminology, an orthodox semigroup is a regular ''E''-semigroup. The term ''orthodox semigroup'' was coined by T. E. Hall and presented in a paper published in 1969. Certain special classes of orthodox semigroups had been studied earlier. For example, semigroups that are also unions of groups, in which the sets of idempotents form subsemigroups were studied by P. H. H. Fantham in 1960.

Examples

*Consider the

binary operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, a binary operation ...

on the set ''S'' = defined by the following

Cayley table Named after the 19th-century United Kingdom, British mathematician Arthur Cayley, a Cayley table describes the structure of a finite group by arranging all the possible products of all the group's elements in a square table reminiscent of an additi ...

: :Then ''S'' is an orthodox semigroup under this operation, the subsemigroup of idempotents being . *

Inverse semigroup In group (mathematics), group theory, an inverse semigroup (occasionally called an inversion semigroup) ''S'' is a semigroup in which every element ''x'' in ''S'' has a unique ''inverse'' ''y'' in ''S'' in the sense that and , i.e. a regular semigr ...

s and bands are examples of orthodox semigroups.

Some elementary properties

The set of idempotents in an orthodox semigroup has several interesting properties. Let ''S'' be a regular semigroup and for any ''a'' in ''S'' let ''V''(''a'') denote the set of inverses of ''a''. Then the following are equivalent: *''S'' is orthodox. *If ''a'' and ''b'' are in ''S'' and if ''x'' is in ''V''(''a'') and ''y'' is in ''V''(''b'') then ''yx'' is in ''V''(''ab''). *If ''e'' is an idempotent in ''S'' then every inverse of ''e'' is also an idempotent. *For every ''a'', ''b'' in ''S'', if ''V''(''a'') ∩ ''V''(''b'') ≠ ∅ then ''V''(''a'') = ''V''(''b'').

Structure

The structure of orthodox semigroups have been determined in terms of bands and inverse semigroups. The Hall–Yamada pullback theorem describes this construction. The construction requires the concepts of

pullback In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward. Precomposition Precomposition with a function probably provides the most elementary notion of pullback: ...

s (in the

category Category, plural categories, may refer to: General uses *Classification, the general act of allocating things to classes/categories Philosophy * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) * Category ( ...

of semigroups) and Nambooripad representation of a fundamental regular semigroup.

References

{{reflist Semigroup theory

Examples

Some elementary properties

Structure

See also

References